THE OPEN ECONOMY SOLOW MODEL: CAPITALMOBILITY
Carl-Johan Dalgaard
Department of Economics
University of Copenhagen
OUTLINE
Part I: Assessing international capital mobility empirically.
— The Feldstein-Horioka Puzzle (S&W-J, Ch. 4.1.)
— The Lucas Paradox (Lucas, 1990)
— Resolving the Lucas paradox? (Caselli and Feyrer, 2005)
Part II: Open economy Solow model - Capital mobility
— The basic model
— Empirical issues
2
PART I
- THE FELDSTEIN-HORIOKA PUZZLE
3
BACKGROUND
In a closed economy setting we know the following must hold
Y = C + I ⇔ I = S.
Hence, total investments (or the investment share of GDP, I/Y ) must
vary 1:1 with total savings (or the savings rate S/Y ).
Thus, a simple regressionµI
Y
¶i= α + β
µS
Y
¶i+ i
should return βOLS = 1 (and αOLS = 0 in the absence of national
accounts mistakes).
4
BACKGROUND
In an open economy, however, things should work differently. In par-
ticular, the following must be true:
S − I = ∆F
If savings exceed domestic investments, the country is building up net
foreign assets. That is, on net the country is investing abroad.
As a result
S −∆F = I. (*)
which says domestic investments equal savings minus what we (on net)
invest abroad.
5
BACKGROUND
Reconsider the regression model from beforeµI
Y
¶i= α + β
µS
Y
¶i+ i
In light of equation (*)
i ≡ −∆F/Y − α.
Estimating the above by OLS we get
βOLS = β −COV
³SY ,∆F/Y
´var
³SY
´as COV
³³SY
´i, i
´= −COV
³SY ,∆F/Y
´. Thus βOLS is expected
to be (much) smaller than 1.
6
THE PUZZLE
The startling finding was, however, this
Figure 1: Source: Feldstein and Horioka, 1980.
Suggests limited capital mobility. In striking contrast to e.g. evidence
on very similar interest rates on similar assets dispite being located in
different countries (thus “a puzzle”).
7
THE PUZZLEThis finding remains something of a puzzle, and is robut to more recent
periods (albeit the size of the coefficient shrinks)
Figure 2: Source: Obstfeld and Rogoff (2000).
To date: No complete resolution.11Perhaps in part because no-one seem to know how small β is supposed to be in order to be consistent with capital mobility.
8
THE LUCAS PARADOX
9
SET-UP
Another contribution striking a similar cord is Lucas (1990).
Lucas’ focus is on rich and poor countries; not just “within the group
of rich”
Basic point of departure is a one good economy, featuring competitive
market.
Firms use a Cobb-Douglas production function. They maximize profits
maxK,L
KαL1−α| {z }=Y
− wL− (r + δ)| {z }user cost of capital
K.
Focusing on FOC wrt K
r + δ = αKα−1L1−α = αkα−1.
Suppose this condition holds in any country.10
THE PARADOXIn particular, suppose we consider India and the US. Then (ignoring δ)
rINDIA = αkα−1INDIA and rUS = αkα−1US .Implying
rINDIA
rUS=
µkINDIA
kUS
¶α−1Capital is hard to measure. But note: y = kα (cf production function).
SOrINDIA
rUS=
µyINDIA
yUS
¶α−1α
Since yUS/yIND ≈ 15 and α = .4, this implies
rINDIA
rUS=
µ1
15
¶.4−1.4≈ 58
Why doesn’t capital flow to poor countries???
11
A SOLUTION TO THE PARADOX?
Maybe we are getting it wrong because we are missing something. Con-
sider the modified production function
Y = XKαL1−α
where X could be human capital (Lucas’ favorit), or something else
(technology). Observe that we now get
y = kαX ⇔ k = (y/X)1/α
Hence the first order condition fromprofit maximization is (still ignoring
δ)
r =MPK = αkα−1X = αyα−1α X
1α
12
A SOLUTION TO THE PARADOX?
Now, ifrIND
rUS≈ 1
then we need
rIND
rUS=
µyIND
yUS
¶α−1αµXIND
XUS
¶ 1α≈ 1
orXus
Xind=
µyusyind
¶1−α= 150.6 ≈ 5.
Lucas manages to motivate “X” almost entirely by human capital; h =
X.
13
WHY IT MAY NOT BE A RESOLUTION1. Evidence for external effects of human capital is not strong.
2. Lucas’ calculation is, under reasonable assumptions, not entirely
internally consistent.
To see this, suppose we rewrite the production function slightly
y = kαX ⇔ y =
µK
Y
¶ α1−α
X11−α
If XusXind
= 5, then³XusXind
´ 11−α
= 511−.4 ≈ 15! If X is human capital, this
implies that we can account for the entire observed difference in labor
productivity by this variable alone (growth “multiplier effect”). Not
plausible.
Of course, things like “A” (TFP) could be included in X. But that
violates the calibration. Another look is warranted.14
A RESOLUTION TO THE LUCAS PARADOX?
15
SET-UP
We begin with a set of basic assumptions
A1 Y = F (K,XL), X= “efficiency" (human capital, productivity).
CRTS: FK ·K + FLL = F = Y
A2 Competitive markets, and multi-good economy (pY 6= pI)
Implication 1R ·K+w ·L = pY Y , where pY is the GDP deflator,
and R is the rental rate of capital (sometimes called: “usercost of
capital", Hall and Jorgenson, 1963)
R = pI · (r + δ) ,
pI = price of investment good.
Implication 2 pY · FK = R and pY · FL = w.
16
SET-UP
Under these assumptions, we can now obtain an estimate for FK. Let
αK ≡ RK/pY Y (capital’s share). Then
FK = αKY
K=
pIpY(r + δ) .
Using data on capital’s share in national accounts, we can calculate FKfor a number of countries.
The question is whether marginal products are equalized ...
17
RESULT 1: MARGINALPRODUCTSARENOTEQUAL-IZED
They are not ...
Figure 3:
18
REFLECTING ON THE RESULT
Investors probably do not care about the marginal product per se. They
care about the return to their investment, r
Perfect capital mobility would require the equalization of the r’s
Fairly easy to calculate the implied r, given the above “view of the
world”:
αKY
K=MPK =
pIpY(r + δ)⇔ r + δ =
pYpI
MPK,
it is assumed that δ is about the same in all countries ...
19
RESULT 2: RENTAL RATES ARE INDEED NOT THATDIFFERENT
Figure 4:
20
SUMMING UP
Lucas: There are no differences in real rates. Human capital is solely re-
sponsible. Shortcoming: Overestimates productivity differences. Ignores
relative price differences
C&F: marginal products are not equalized. But rental rates are not
that different.
— Rich places have a lot of capital (even in a fully integrated world)
because: “X” is large (not only human capital though), and because
the relative price of investment is low (pI/py).
— Hence if international capital markets do allocate capital reasonably
efficiently, then we better think about how it affects our understanding
of the growth process
21
Part II: OPEN ECONOMY SOLOWMODEL
- CAPITAL MOBILITY
22
THE BASIC MODEL: SET UP
We are considering an open economy, where capital is fully mobile.
Labor, however, is not. All markets are competitive.
Two new basic relationsships:
1. Savings 6= Domestic total investment
2. Production and Income are not longer identical
The national accounts identity is
Y = C + I +NX ⇐⇒ Y + rF = C + I +NX + rF
whereNX represents net exports, and F is holdnings of foreign capital;
rF=income inflow from foreign capital holdnings.
23
THE BASIC MODEL: SET UP
Gross National Income (GNI) is therefore Y +rF , cf 2. Y is production,
or, Gross Domestic Product.
Note that if NX>0 the economy must be building up assets abroad
(exports > imports). Hence, in general (r is assumed constant):
NXt + rF = Ft+1 − Ft
Finally, by definition
St = Yt + rFt − Ct
Combining
St = It + Ft+1 − Ft
Hence, savings can be used to accumulate domestic capital (I), or,
foreign assets (Ft+1 − Ft), cf. 1.
24
THE BASIC MODEL: SET UP
As “usual” we have that
Kt+1 = It +Kt
(i.e., here we assume δ = 0, for simplicity)
But, observe that we now have
Kt+1 = St − (Ft+1 − Ft) +Kt⇔ Kt+1 + Ft+1 = St +Kt + Ft
If we define total wealth (domestically owned local (K) and Foreign (F )
capital)
Vt = Kt + Ft
Leaving us with
Vt+1 = St + Vt. (1)
25
THE BASIC MODEL: SET UP
The fundamental assumption about savings behavior is the same: Peo-
ple save a constant fraction of total income. In the open economy
St = s · (Yt + rFt) , 0 < s < 1. (2)
We will also maintain our basic assumption about production
Yt = F (Kt, Lt;A) = AKαt L
1−at
From (A) competitive market, and (B) constant returns to scale follows
Yt = wtLt + rKt (3)
Since
wt =∂F [·]∂Lt
, r =∂F [·]∂Kt
26
THE BASIC MODEL: SET UP
The fact that capital is fully mobile has an important implication. De-
note by rw the world real rate of interest. Then at all points in time
rw = r =∂F [·]∂Kt
Substituting for ∂F [·]∂Ktwe find
rw = αAkα−1⇔ k =
µαA
rw
¶ 11−α
.
Hence the capital-labor ratio is constant, absent changes in A. Suppose
A rises ...
Note also, that this implies a constant wage rate
wt = w =∂F [·]∂Lt
= (1− α)Akα (4)
27
SOLVING THE BASIC MODEL
Starting with eq (1):
Vt+1 = St + Vteq (2)= s · (Yt + rwFt) + Vt
eq (3)= s · (wLt + rwKt + rwFt) + Vt
def of V and eq (4)= s · wLt + (1 + srw)Vt
As a final step: Lt+1 = (1 + n)Lt, n > −1. Let vt ≡ Vt/Lt. Then
vt+1 =sw
1 + n+1 + srw
1 + nvt ≡ Φ (vt)
is the fundamental law of motion for wealth in the open economy setting.
[Insert Phasediagram]
28
SOLVING THE BASIC MODEL
DefinitionA steady state of the model is a vt+1 = vt = v∗ such thatv∗ = Φ (v)
For existence of a steady state, we require the following stability con-
dition
Φ0 (v) =1 + srw
1 + n< 1⇔ srw < n
Plausible? r is the world market interest rate. The “world” is a closed
economy. In the steady state of a closed economy the real rate is given
by (when δ = g = 0)
r∗ =MPK∗ = α
µY
K
¶∗= α
n
s.
For the world, define rw = αnw/sw. Inserted
s
µαnw
sw
¶< n⇔ α
s
n<
sw
nw.
29
SOME OBSERVATIONS ABOUT THE STEADY STATEUnique (non-trivial) steady state, where
v∗ =sw
n− srw,
w = (1− α) y = (1− α)A1/(1−α) (α/rw)α1−α
Globally stable. For any v0 > 0 limt→∞ vt→ v∗
v∗ determined by local structural charactaristics: s,A, n.
Specifically: ∂v∗/∂s > 0, ∂v∗/∂n < 0 and ∂v∗/∂A > 0.
Qualitatively, just as in a closed economy Solow model (with k ex-
changed for v).
30
SOME OBSERVATIONS ABOUT THE STEADY STATENet foreign position?
Rercall that v∗ = k + f∗. If we use that rw = αAkα−1 and thatw = (1− α)Akα, we obtain that
rw
w=
α
(1− α)
kα−1
kα⇔ k =
α
1− α
w
rw.
As a result
f∗ = v∗ − k =sw
n− sr− α
1− α
w
rw=
1
1− α
s
n
1
rw
∙rw − αn/s
1− srw/n
¸w.
Recall, the steady state autarky real rate of return r∗ = αn/s. Hence,
if rw > r∗ ⇒ creditor, and debitor otherwise (rw < r∗).
Of course, a “low” r∗ implies high savings, and vice versa.
31
EMPIRICS: LONG-RUN GROWTH
Observe that GNI per capita
ynt = y + rft = w + rwvt
We can therefore convert the law of motion for v, into one for yn
1
r
¡ynt+1 − w
¢=
sw
1 + n+1 + sr
1 + n
µ1
r(ynt − w)
¶⇓
ynt+1 =n
1 + nw +
1 + sr
1 + nynt ⇒ (yn)∗ =
n
n− srw.
As in the closed economy Solow model: Growth in income per capita
will come to a halt (provided 1+sr1+n < 1).
You will still see growth in transition however. Note, GDP per capita,
y (as well as w, r and k), is constant at all points in time. Changes
require human capital accumulation, or, technological change.32
EMPIRICS: CONVERGENCE PROCESS
Unique steady state⇒ Model predicts conditional convergence.
Since n, rw and s are constants, we can solve for the entire path for yn
ynt+1 =n
1 + nw+
1 + srw
1 + nynt ⇔ ynt =
µ1 + srw
1 + n
¶t ¡yn0 − (yn)
∗¢+(yn)∗with (yn)∗ ≡ nw
n−srw.
Speed of convergence?
ynt − (yn)∗
yn0 − (yn)∗ =
1
2=
µ1 + srw
1 + n
¶t1/2⇒ t1/2 =
− log 2log³1+srw1+n
´..
33
EMPIRICS: CONVERGENCE PROCESS
In the closed economy (if δ = g = 0)
tc1/2 =− ln (2)ln³1+αn1+n
´Hence by comparison
tc1/2 < topen1/2⇔ − ln (2)ln³1+αn1+n
´ <− log 2
log³1+srw1+n
´ ⇔ rw > αn
s= r∗
Hence slower if a creditor ( rw > r∗), and vice versa for a debitor.
Intuition...
34
EMPIRICS: INCOME DIFFERENCES
If we compare two countries who only differ in terms of savings (1 and
2, respectively)yn1yn2=
nn−s1rwn
n−s2rw=n− s2r
w
n− s1rw
Using reasonable parameter values (rw = 0.03 and population growth
in the world of about 2 percent), maximum variation in savings rates
translate into:n− s2r
w
n− s1rw=0.02− 0.1 · 0.030.02− 0.4 · 0.03 ≈ 2,
which is about the same income difference that we could generate in
the closed economy Solow model, with similar (1:4) variation in s.
35
SUMMING UPThe open economy model does not radically change our priors viz stan-
dard model.
Countries that save more, have slower population growth and higher
levels of technological sophistication are still predicted to the more pros-
perious.
No growth in the long-run (absent...). Lengthy transitions can be mo-
tivated
The model does about as well as Solow model in accounting for per
capita income differences
But we can ask new questions: Does liberalising capital mobility in-
crease income per capita? How will a world wide credit-crunch (higher
rw) impact living standards?
36
ADVANTAGES OF FREE CAPITAL MOBILITY?Compare levels of national income per capita in the two settings (open
vs closed). Closed: attain a steady state associated with r∗.
RESULT: We find national income always rises, unless rw = r∗. For-mal proof p. 113-14
Suppose rw > r∗. In stead of being forced to invest at home, at therate r∗, the country can now invest abrod, and reap the gains rw > r.
This will increase national income. Impact on wages?
Suppose rw < r∗. In the absence of capital mobility the economy wouldhave ended up in a low steady state (note: r∗ is high). Opening up tocapital flows therefore leads to capital imports which increases GDP per
capita and therefore GNI. Impact on wages?
37
CHANGES IN THE REAL RATE OF INTEREST
Permanent increase in rw.
When rw > r ⇒ k ↓⇒ y ↓→ yn ↓In addition: rw ↑⇒ rwf ↑ (if f > 0)⇒ yn ↑Creditor: Stands to win in terms of income; due to the second mecha-
nism.
Debitor: Stands to loose in terms of income.
In either case, capital flows out, and depresses wages
An example of how international capital market can lead to fluctuations
in income.
38